\(\int \frac {\csc ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) [202]

Optimal result

Integrand size = 24, antiderivative size = 229 \[ \int \frac {\csc ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {b^{5/4} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {(3 a+8 b) \text {arctanh}(\cos (c+d x))}{8 a^2 d}+\frac {b^{5/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {1}{16 a d (1-\cos (c+d x))^2}-\frac {3}{16 a d (1-\cos (c+d x))}+\frac {1}{16 a d (1+\cos (c+d x))^2}+\frac {3}{16 a d (1+\cos (c+d x))} \]

[Out]

-1/8*(3*a+8*b)*arctanh(cos(d*x+c))/a^2/d-1/16/a/d/(1-cos(d*x+c))^2-3/16/a/d/(1-cos(d*x+c))+1/16/a/d/(1+cos(d*x
+c))^2+3/16/a/d/(1+cos(d*x+c))-1/2*b^(5/4)*arctan(b^(1/4)*cos(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))/a^2/d/(a^(1/2)-b
^(1/2))^(1/2)+1/2*b^(5/4)*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))/a^2/d/(a^(1/2)+b^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3294, 1184, 213, 1180, 211, 214} \[ \int \frac {\csc ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {b^{5/4} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^2 d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {b^{5/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^2 d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {(3 a+8 b) \text {arctanh}(\cos (c+d x))}{8 a^2 d}-\frac {3}{16 a d (1-\cos (c+d x))}+\frac {3}{16 a d (\cos (c+d x)+1)}-\frac {1}{16 a d (1-\cos (c+d x))^2}+\frac {1}{16 a d (\cos (c+d x)+1)^2} \]

[In]

Int[Csc[c + d*x]^5/(a - b*Sin[c + d*x]^4),x]

[Out]

-1/2*(b^(5/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(a^2*Sqrt[Sqrt[a] - Sqrt[b]]*d) - ((3*a
+ 8*b)*ArcTanh[Cos[c + d*x]])/(8*a^2*d) + (b^(5/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2
*a^2*Sqrt[Sqrt[a] + Sqrt[b]]*d) - 1/(16*a*d*(1 - Cos[c + d*x])^2) - 3/(16*a*d*(1 - Cos[c + d*x])) + 1/(16*a*d*
(1 + Cos[c + d*x])^2) + 3/(16*a*d*(1 + Cos[c + d*x]))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1184

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x
^2)^q/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a
*e^2, 0] && IntegerQ[q]

Rule 3294

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^3 \left (a-b+2 b x^2-b x^4\right )} \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \left (-\frac {1}{8 a (-1+x)^3}+\frac {3}{16 a (-1+x)^2}+\frac {1}{8 a (1+x)^3}+\frac {3}{16 a (1+x)^2}+\frac {-3 a-8 b}{8 a^2 \left (-1+x^2\right )}-\frac {b^2 \left (-1+x^2\right )}{a^2 \left (a-b+2 b x^2-b x^4\right )}\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {1}{16 a d (1-\cos (c+d x))^2}-\frac {3}{16 a d (1-\cos (c+d x))}+\frac {1}{16 a d (1+\cos (c+d x))^2}+\frac {3}{16 a d (1+\cos (c+d x))}+\frac {b^2 \text {Subst}\left (\int \frac {-1+x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{a^2 d}+\frac {(3 a+8 b) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\cos (c+d x)\right )}{8 a^2 d} \\ & = -\frac {(3 a+8 b) \text {arctanh}(\cos (c+d x))}{8 a^2 d}-\frac {1}{16 a d (1-\cos (c+d x))^2}-\frac {3}{16 a d (1-\cos (c+d x))}+\frac {1}{16 a d (1+\cos (c+d x))^2}+\frac {3}{16 a d (1+\cos (c+d x))}+\frac {b^2 \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 a^2 d}+\frac {b^2 \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 a^2 d} \\ & = -\frac {b^{5/4} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {(3 a+8 b) \text {arctanh}(\cos (c+d x))}{8 a^2 d}+\frac {b^{5/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {1}{16 a d (1-\cos (c+d x))^2}-\frac {3}{16 a d (1-\cos (c+d x))}+\frac {1}{16 a d (1+\cos (c+d x))^2}+\frac {3}{16 a d (1+\cos (c+d x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 1.33 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.79 \[ \int \frac {\csc ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {-6 a \csc ^2\left (\frac {1}{2} (c+d x)\right )-a \csc ^4\left (\frac {1}{2} (c+d x)\right )-24 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-64 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+24 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+64 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-8 i b^2 \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+6 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-3 i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-6 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+3 i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]+6 a \sec ^2\left (\frac {1}{2} (c+d x)\right )+a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 a^2 d} \]

[In]

Integrate[Csc[c + d*x]^5/(a - b*Sin[c + d*x]^4),x]

[Out]

(-6*a*Csc[(c + d*x)/2]^2 - a*Csc[(c + d*x)/2]^4 - 24*a*Log[Cos[(c + d*x)/2]] - 64*b*Log[Cos[(c + d*x)/2]] + 24
*a*Log[Sin[(c + d*x)/2]] + 64*b*Log[Sin[(c + d*x)/2]] - (8*I)*b^2*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4
- 4*b*#1^6 + b*#1^8 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 6
*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (3*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - 6*ArcTan[Sin[c
+ d*x]/(Cos[c + d*x] - #1)]*#1^4 + (3*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + 2*ArcTan[Sin[c + d*x]/(Cos[
c + d*x] - #1)]*#1^6 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b
*#1^7) & ] + 6*a*Sec[(c + d*x)/2]^2 + a*Sec[(c + d*x)/2]^4)/(64*a^2*d)

Maple [A] (verified)

Time = 1.93 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.84

method	result	size
derivativedivides	\(\frac {\frac {1}{16 a \left (1+\cos \left (d x +c \right )\right )^{2}}+\frac {3}{16 a \left (1+\cos \left (d x +c \right )\right )}+\frac {\left (-3 a -8 b \right ) \ln \left (1+\cos \left (d x +c \right )\right )}{16 a^{2}}-\frac {b^{3} \left (\frac {\arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 b \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 b \sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{a^{2}}-\frac {1}{16 a \left (\cos \left (d x +c \right )-1\right )^{2}}+\frac {3}{16 a \left (\cos \left (d x +c \right )-1\right )}+\frac {\left (3 a +8 b \right ) \ln \left (\cos \left (d x +c \right )-1\right )}{16 a^{2}}}{d}\)	\(193\)
default	\(\frac {\frac {1}{16 a \left (1+\cos \left (d x +c \right )\right )^{2}}+\frac {3}{16 a \left (1+\cos \left (d x +c \right )\right )}+\frac {\left (-3 a -8 b \right ) \ln \left (1+\cos \left (d x +c \right )\right )}{16 a^{2}}-\frac {b^{3} \left (\frac {\arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 b \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 b \sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{a^{2}}-\frac {1}{16 a \left (\cos \left (d x +c \right )-1\right )^{2}}+\frac {3}{16 a \left (\cos \left (d x +c \right )-1\right )}+\frac {\left (3 a +8 b \right ) \ln \left (\cos \left (d x +c \right )-1\right )}{16 a^{2}}}{d}\)	\(193\)
risch	\(\frac {3 \,{\mathrm e}^{7 i \left (d x +c \right )}-11 \,{\mathrm e}^{5 i \left (d x +c \right )}-11 \,{\mathrm e}^{3 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}}{4 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d a}+\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{2} d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d a}-\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{2} d}+32 i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (268435456 a^{9} d^{4}-268435456 a^{8} b \,d^{4}\right ) \textit {\_Z}^{4}-32768 a^{4} b^{3} d^{2} \textit {\_Z}^{2}-b^{5}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (-\frac {4194304 i d^{3} a^{7}}{b^{4}}+\frac {4194304 i a^{6} d^{3}}{b^{3}}\right ) \textit {\_R}^{3}+\frac {512 i a^{2} d \textit {\_R}}{b}\right ) {\mathrm e}^{i \left (d x +c \right )}+1\right )\right )\)	\(263\)

[In]

int(csc(d*x+c)^5/(a-b*sin(d*x+c)^4),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/16/a/(1+cos(d*x+c))^2+3/16/a/(1+cos(d*x+c))+1/16/a^2*(-3*a-8*b)*ln(1+cos(d*x+c))-b^3/a^2*(1/2/b/(((a*b)
^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))-1/2/b/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(cos(
d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2)))-1/16/a/(cos(d*x+c)-1)^2+3/16/a/(cos(d*x+c)-1)+1/16*(3*a+8*b)/a^2*ln(cos(d
*x+c)-1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1089 vs. \(2 (179) = 358\).

Time = 0.43 (sec) , antiderivative size = 1089, normalized size of antiderivative = 4.76 \[ \int \frac {\csc ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(csc(d*x+c)^5/(a-b*sin(d*x+c)^4),x, algorithm="fricas")

[Out]

1/16*(6*a*cos(d*x + c)^3 + 4*(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)*sqrt(-((a^5 - a^4*b)*d^2*
sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^3)/((a^5 - a^4*b)*d^2))*log(b^4*cos(d*x + c) + (a^2*b^3*d - (a^7
- a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)))*sqrt(-((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a
^7*b^2)*d^4)) + b^3)/((a^5 - a^4*b)*d^2))) - 4*(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)*sqrt(((
a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^3)/((a^5 - a^4*b)*d^2))*log(b^4*cos(d*x + c) -
(a^2*b^3*d + (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)))*sqrt(((a^5 - a^4*b)*d^2*sqrt(b^5/((a
^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^3)/((a^5 - a^4*b)*d^2))) - 4*(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2
+ a^2*d)*sqrt(-((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^3)/((a^5 - a^4*b)*d^2))*log(-b
^4*cos(d*x + c) + (a^2*b^3*d - (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)))*sqrt(-((a^5 - a^4*
b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^3)/((a^5 - a^4*b)*d^2))) + 4*(a^2*d*cos(d*x + c)^4 - 2*a^
2*d*cos(d*x + c)^2 + a^2*d)*sqrt(((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^3)/((a^5 - a
^4*b)*d^2))*log(-b^4*cos(d*x + c) - (a^2*b^3*d + (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)))*
sqrt(((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^3)/((a^5 - a^4*b)*d^2))) - 10*a*cos(d*x
+ c) - ((3*a + 8*b)*cos(d*x + c)^4 - 2*(3*a + 8*b)*cos(d*x + c)^2 + 3*a + 8*b)*log(1/2*cos(d*x + c) + 1/2) + (
(3*a + 8*b)*cos(d*x + c)^4 - 2*(3*a + 8*b)*cos(d*x + c)^2 + 3*a + 8*b)*log(-1/2*cos(d*x + c) + 1/2))/(a^2*d*co
s(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)

Sympy [F]

\[ \int \frac {\csc ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int \frac {\csc ^{5}{\left (c + d x \right )}}{a - b \sin ^{4}{\left (c + d x \right )}}\, dx \]

[In]

integrate(csc(d*x+c)**5/(a-b*sin(d*x+c)**4),x)

[Out]

Integral(csc(c + d*x)**5/(a - b*sin(c + d*x)**4), x)

Maxima [F]

\[ \int \frac {\csc ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\csc \left (d x + c\right )^{5}}{b \sin \left (d x + c\right )^{4} - a} \,d x } \]

[In]

integrate(csc(d*x+c)^5/(a-b*sin(d*x+c)^4),x, algorithm="maxima")

[Out]

-1/16*(48*a*cos(2*d*x + 2*c)*cos(d*x + c) - 176*a*sin(3*d*x + 3*c)*sin(2*d*x + 2*c) + 48*a*sin(2*d*x + 2*c)*si
n(d*x + c) - 4*(3*a*cos(7*d*x + 7*c) - 11*a*cos(5*d*x + 5*c) - 11*a*cos(3*d*x + 3*c) + 3*a*cos(d*x + c))*cos(8
*d*x + 8*c) + 12*(4*a*cos(6*d*x + 6*c) - 6*a*cos(4*d*x + 4*c) + 4*a*cos(2*d*x + 2*c) - a)*cos(7*d*x + 7*c) - 1
6*(11*a*cos(5*d*x + 5*c) + 11*a*cos(3*d*x + 3*c) - 3*a*cos(d*x + c))*cos(6*d*x + 6*c) + 44*(6*a*cos(4*d*x + 4*
c) - 4*a*cos(2*d*x + 2*c) + a)*cos(5*d*x + 5*c) + 24*(11*a*cos(3*d*x + 3*c) - 3*a*cos(d*x + c))*cos(4*d*x + 4*
c) - 44*(4*a*cos(2*d*x + 2*c) - a)*cos(3*d*x + 3*c) - 12*a*cos(d*x + c) + 16*(a^2*d*cos(8*d*x + 8*c)^2 + 16*a^
2*d*cos(6*d*x + 6*c)^2 + 36*a^2*d*cos(4*d*x + 4*c)^2 + 16*a^2*d*cos(2*d*x + 2*c)^2 + a^2*d*sin(8*d*x + 8*c)^2
+ 16*a^2*d*sin(6*d*x + 6*c)^2 + 36*a^2*d*sin(4*d*x + 4*c)^2 - 48*a^2*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*
a^2*d*sin(2*d*x + 2*c)^2 - 8*a^2*d*cos(2*d*x + 2*c) + a^2*d - 2*(4*a^2*d*cos(6*d*x + 6*c) - 6*a^2*d*cos(4*d*x
+ 4*c) + 4*a^2*d*cos(2*d*x + 2*c) - a^2*d)*cos(8*d*x + 8*c) - 8*(6*a^2*d*cos(4*d*x + 4*c) - 4*a^2*d*cos(2*d*x
+ 2*c) + a^2*d)*cos(6*d*x + 6*c) - 12*(4*a^2*d*cos(2*d*x + 2*c) - a^2*d)*cos(4*d*x + 4*c) - 4*(2*a^2*d*sin(6*d
*x + 6*c) - 3*a^2*d*sin(4*d*x + 4*c) + 2*a^2*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 16*(3*a^2*d*sin(4*d*x + 4*
c) - 2*a^2*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*integrate(-2*(12*b^3*cos(3*d*x + 3*c)*sin(2*d*x + 2*c) - 4*b^
3*cos(d*x + c)*sin(2*d*x + 2*c) + 4*b^3*cos(2*d*x + 2*c)*sin(d*x + c) - b^3*sin(d*x + c) + (b^3*sin(7*d*x + 7*
c) - 3*b^3*sin(5*d*x + 5*c) + 3*b^3*sin(3*d*x + 3*c) - b^3*sin(d*x + c))*cos(8*d*x + 8*c) + 2*(2*b^3*sin(6*d*x
+ 6*c) + 2*b^3*sin(2*d*x + 2*c) + (8*a*b^2 - 3*b^3)*sin(4*d*x + 4*c))*cos(7*d*x + 7*c) + 4*(3*b^3*sin(5*d*x +
5*c) - 3*b^3*sin(3*d*x + 3*c) + b^3*sin(d*x + c))*cos(6*d*x + 6*c) - 6*(2*b^3*sin(2*d*x + 2*c) + (8*a*b^2 - 3
*b^3)*sin(4*d*x + 4*c))*cos(5*d*x + 5*c) - 2*(3*(8*a*b^2 - 3*b^3)*sin(3*d*x + 3*c) - (8*a*b^2 - 3*b^3)*sin(d*x
+ c))*cos(4*d*x + 4*c) - (b^3*cos(7*d*x + 7*c) - 3*b^3*cos(5*d*x + 5*c) + 3*b^3*cos(3*d*x + 3*c) - b^3*cos(d*
x + c))*sin(8*d*x + 8*c) - (4*b^3*cos(6*d*x + 6*c) + 4*b^3*cos(2*d*x + 2*c) - b^3 + 2*(8*a*b^2 - 3*b^3)*cos(4*
d*x + 4*c))*sin(7*d*x + 7*c) - 4*(3*b^3*cos(5*d*x + 5*c) - 3*b^3*cos(3*d*x + 3*c) + b^3*cos(d*x + c))*sin(6*d*
x + 6*c) + 3*(4*b^3*cos(2*d*x + 2*c) - b^3 + 2*(8*a*b^2 - 3*b^3)*cos(4*d*x + 4*c))*sin(5*d*x + 5*c) + 2*(3*(8*
a*b^2 - 3*b^3)*cos(3*d*x + 3*c) - (8*a*b^2 - 3*b^3)*cos(d*x + c))*sin(4*d*x + 4*c) - 3*(4*b^3*cos(2*d*x + 2*c)
- b^3)*sin(3*d*x + 3*c))/(a^2*b^2*cos(8*d*x + 8*c)^2 + 16*a^2*b^2*cos(6*d*x + 6*c)^2 + 16*a^2*b^2*cos(2*d*x +
2*c)^2 + a^2*b^2*sin(8*d*x + 8*c)^2 + 16*a^2*b^2*sin(6*d*x + 6*c)^2 + 16*a^2*b^2*sin(2*d*x + 2*c)^2 - 8*a^2*b
^2*cos(2*d*x + 2*c) + a^2*b^2 + 4*(64*a^4 - 48*a^3*b + 9*a^2*b^2)*cos(4*d*x + 4*c)^2 + 4*(64*a^4 - 48*a^3*b +
9*a^2*b^2)*sin(4*d*x + 4*c)^2 + 16*(8*a^3*b - 3*a^2*b^2)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - 2*(4*a^2*b^2*cos(
6*d*x + 6*c) + 4*a^2*b^2*cos(2*d*x + 2*c) - a^2*b^2 + 2*(8*a^3*b - 3*a^2*b^2)*cos(4*d*x + 4*c))*cos(8*d*x + 8*
c) + 8*(4*a^2*b^2*cos(2*d*x + 2*c) - a^2*b^2 + 2*(8*a^3*b - 3*a^2*b^2)*cos(4*d*x + 4*c))*cos(6*d*x + 6*c) - 4*
(8*a^3*b - 3*a^2*b^2 - 4*(8*a^3*b - 3*a^2*b^2)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 4*(2*a^2*b^2*sin(6*d*x + 6
*c) + 2*a^2*b^2*sin(2*d*x + 2*c) + (8*a^3*b - 3*a^2*b^2)*sin(4*d*x + 4*c))*sin(8*d*x + 8*c) + 16*(2*a^2*b^2*si
n(2*d*x + 2*c) + (8*a^3*b - 3*a^2*b^2)*sin(4*d*x + 4*c))*sin(6*d*x + 6*c)), x) + ((3*a + 8*b)*cos(8*d*x + 8*c)
^2 + 16*(3*a + 8*b)*cos(6*d*x + 6*c)^2 + 36*(3*a + 8*b)*cos(4*d*x + 4*c)^2 + 16*(3*a + 8*b)*cos(2*d*x + 2*c)^2
+ (3*a + 8*b)*sin(8*d*x + 8*c)^2 + 16*(3*a + 8*b)*sin(6*d*x + 6*c)^2 + 36*(3*a + 8*b)*sin(4*d*x + 4*c)^2 - 48
*(3*a + 8*b)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(3*a + 8*b)*sin(2*d*x + 2*c)^2 - 2*(4*(3*a + 8*b)*cos(6*d*
x + 6*c) - 6*(3*a + 8*b)*cos(4*d*x + 4*c) + 4*(3*a + 8*b)*cos(2*d*x + 2*c) - 3*a - 8*b)*cos(8*d*x + 8*c) - 8*(
6*(3*a + 8*b)*cos(4*d*x + 4*c) - 4*(3*a + 8*b)*cos(2*d*x + 2*c) + 3*a + 8*b)*cos(6*d*x + 6*c) - 12*(4*(3*a + 8
*b)*cos(2*d*x + 2*c) - 3*a - 8*b)*cos(4*d*x + 4*c) - 8*(3*a + 8*b)*cos(2*d*x + 2*c) - 4*(2*(3*a + 8*b)*sin(6*d
*x + 6*c) - 3*(3*a + 8*b)*sin(4*d*x + 4*c) + 2*(3*a + 8*b)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 16*(3*(3*a + 8
*b)*sin(4*d*x + 4*c) - 2*(3*a + 8*b)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 3*a + 8*b)*log(cos(d*x)^2 + 2*cos(d*
x)*cos(c) + cos(c)^2 + sin(d*x)^2 - 2*sin(d*x)*sin(c) + sin(c)^2) - ((3*a + 8*b)*cos(8*d*x + 8*c)^2 + 16*(3*a
+ 8*b)*cos(6*d*x + 6*c)^2 + 36*(3*a + 8*b)*cos(4*d*x + 4*c)^2 + 16*(3*a + 8*b)*cos(2*d*x + 2*c)^2 + (3*a + 8*b
)*sin(8*d*x + 8*c)^2 + 16*(3*a + 8*b)*sin(6*d*x + 6*c)^2 + 36*(3*a + 8*b)*sin(4*d*x + 4*c)^2 - 48*(3*a + 8*b)*
sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(3*a + 8*b)*sin(2*d*x + 2*c)^2 - 2*(4*(3*a + 8*b)*cos(6*d*x + 6*c) - 6*
(3*a + 8*b)*cos(4*d*x + 4*c) + 4*(3*a + 8*b)*cos(2*d*x + 2*c) - 3*a - 8*b)*cos(8*d*x + 8*c) - 8*(6*(3*a + 8*b)
*cos(4*d*x + 4*c) - 4*(3*a + 8*b)*cos(2*d*x + 2*c) + 3*a + 8*b)*cos(6*d*x + 6*c) - 12*(4*(3*a + 8*b)*cos(2*d*x
+ 2*c) - 3*a - 8*b)*cos(4*d*x + 4*c) - 8*(3*a + 8*b)*cos(2*d*x + 2*c) - 4*(2*(3*a + 8*b)*sin(6*d*x + 6*c) - 3
*(3*a + 8*b)*sin(4*d*x + 4*c) + 2*(3*a + 8*b)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 16*(3*(3*a + 8*b)*sin(4*d*x
+ 4*c) - 2*(3*a + 8*b)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 3*a + 8*b)*log(cos(d*x)^2 - 2*cos(d*x)*cos(c) + c
os(c)^2 + sin(d*x)^2 + 2*sin(d*x)*sin(c) + sin(c)^2) - 4*(3*a*sin(7*d*x + 7*c) - 11*a*sin(5*d*x + 5*c) - 11*a*
sin(3*d*x + 3*c) + 3*a*sin(d*x + c))*sin(8*d*x + 8*c) + 24*(2*a*sin(6*d*x + 6*c) - 3*a*sin(4*d*x + 4*c) + 2*a*
sin(2*d*x + 2*c))*sin(7*d*x + 7*c) - 16*(11*a*sin(5*d*x + 5*c) + 11*a*sin(3*d*x + 3*c) - 3*a*sin(d*x + c))*sin
(6*d*x + 6*c) + 88*(3*a*sin(4*d*x + 4*c) - 2*a*sin(2*d*x + 2*c))*sin(5*d*x + 5*c) + 24*(11*a*sin(3*d*x + 3*c)
- 3*a*sin(d*x + c))*sin(4*d*x + 4*c))/(a^2*d*cos(8*d*x + 8*c)^2 + 16*a^2*d*cos(6*d*x + 6*c)^2 + 36*a^2*d*cos(4
*d*x + 4*c)^2 + 16*a^2*d*cos(2*d*x + 2*c)^2 + a^2*d*sin(8*d*x + 8*c)^2 + 16*a^2*d*sin(6*d*x + 6*c)^2 + 36*a^2*
d*sin(4*d*x + 4*c)^2 - 48*a^2*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*a^2*d*sin(2*d*x + 2*c)^2 - 8*a^2*d*cos(
2*d*x + 2*c) + a^2*d - 2*(4*a^2*d*cos(6*d*x + 6*c) - 6*a^2*d*cos(4*d*x + 4*c) + 4*a^2*d*cos(2*d*x + 2*c) - a^2
*d)*cos(8*d*x + 8*c) - 8*(6*a^2*d*cos(4*d*x + 4*c) - 4*a^2*d*cos(2*d*x + 2*c) + a^2*d)*cos(6*d*x + 6*c) - 12*(
4*a^2*d*cos(2*d*x + 2*c) - a^2*d)*cos(4*d*x + 4*c) - 4*(2*a^2*d*sin(6*d*x + 6*c) - 3*a^2*d*sin(4*d*x + 4*c) +
2*a^2*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 16*(3*a^2*d*sin(4*d*x + 4*c) - 2*a^2*d*sin(2*d*x + 2*c))*sin(6*d*
x + 6*c))

Giac [F]

\[ \int \frac {\csc ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\csc \left (d x + c\right )^{5}}{b \sin \left (d x + c\right )^{4} - a} \,d x } \]

[In]

integrate(csc(d*x+c)^5/(a-b*sin(d*x+c)^4),x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 15.24 (sec) , antiderivative size = 3692, normalized size of antiderivative = 16.12 \[ \int \frac {\csc ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]

[In]

int(1/(sin(c + d*x)^5*(a - b*sin(c + d*x)^4)),x)

[Out]

(atan(((((768*a^3*b^8 - 144*a^5*b^6)/(64*a^5) + (((10240*a^8*b^5 - 12288*a^7*b^6 + 6144*a^9*b^4)/(64*a^5) - (c
os(c + d*x)*(12288*a^8*b^5 - 8192*a^9*b^4)*(((a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))^(1/2))/(16*a^4))*(
((a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))^(1/2) + (cos(c + d*x)*(2304*a^4*b^7 + 768*a^5*b^6 + 144*a^6*b^
5))/(16*a^4))*(((a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))^(1/2))*(((a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b
- a^9)))^(1/2) - (cos(c + d*x)*(48*a*b^8 + 96*b^9 + 9*a^2*b^7))/(16*a^4))*(((a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^
8*b - a^9)))^(1/2)*1i - (((768*a^3*b^8 - 144*a^5*b^6)/(64*a^5) + (((10240*a^8*b^5 - 12288*a^7*b^6 + 6144*a^9*b
^4)/(64*a^5) + (cos(c + d*x)*(12288*a^8*b^5 - 8192*a^9*b^4)*(((a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))^(
1/2))/(16*a^4))*(((a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))^(1/2) - (cos(c + d*x)*(2304*a^4*b^7 + 768*a^5
*b^6 + 144*a^6*b^5))/(16*a^4))*(((a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))^(1/2))*(((a^9*b^5)^(1/2) + a^4
*b^3)/(16*(a^8*b - a^9)))^(1/2) + (cos(c + d*x)*(48*a*b^8 + 96*b^9 + 9*a^2*b^7))/(16*a^4))*(((a^9*b^5)^(1/2) +
a^4*b^3)/(16*(a^8*b - a^9)))^(1/2)*1i)/((((768*a^3*b^8 - 144*a^5*b^6)/(64*a^5) + (((10240*a^8*b^5 - 12288*a^7
*b^6 + 6144*a^9*b^4)/(64*a^5) - (cos(c + d*x)*(12288*a^8*b^5 - 8192*a^9*b^4)*(((a^9*b^5)^(1/2) + a^4*b^3)/(16*
(a^8*b - a^9)))^(1/2))/(16*a^4))*(((a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))^(1/2) + (cos(c + d*x)*(2304*
a^4*b^7 + 768*a^5*b^6 + 144*a^6*b^5))/(16*a^4))*(((a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))^(1/2))*(((a^9
*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))^(1/2) - (cos(c + d*x)*(48*a*b^8 + 96*b^9 + 9*a^2*b^7))/(16*a^4))*((
(a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))^(1/2) + (((768*a^3*b^8 - 144*a^5*b^6)/(64*a^5) + (((10240*a^8*b
^5 - 12288*a^7*b^6 + 6144*a^9*b^4)/(64*a^5) + (cos(c + d*x)*(12288*a^8*b^5 - 8192*a^9*b^4)*(((a^9*b^5)^(1/2) +
a^4*b^3)/(16*(a^8*b - a^9)))^(1/2))/(16*a^4))*(((a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))^(1/2) - (cos(c
+ d*x)*(2304*a^4*b^7 + 768*a^5*b^6 + 144*a^6*b^5))/(16*a^4))*(((a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))
^(1/2))*(((a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))^(1/2) + (cos(c + d*x)*(48*a*b^8 + 96*b^9 + 9*a^2*b^7)
)/(16*a^4))*(((a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))^(1/2) + (9*a*b^8 + 24*b^9)/(32*a^5)))*(((a^9*b^5)
^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))^(1/2)*2i)/d + (atan(((((768*a^3*b^8 - 144*a^5*b^6)/(64*a^5) + (((10240*a
^8*b^5 - 12288*a^7*b^6 + 6144*a^9*b^4)/(64*a^5) - (cos(c + d*x)*(12288*a^8*b^5 - 8192*a^9*b^4)*(-((a^9*b^5)^(1
/2) - a^4*b^3)/(16*(a^8*b - a^9)))^(1/2))/(16*a^4))*(-((a^9*b^5)^(1/2) - a^4*b^3)/(16*(a^8*b - a^9)))^(1/2) +
(cos(c + d*x)*(2304*a^4*b^7 + 768*a^5*b^6 + 144*a^6*b^5))/(16*a^4))*(-((a^9*b^5)^(1/2) - a^4*b^3)/(16*(a^8*b -
a^9)))^(1/2))*(-((a^9*b^5)^(1/2) - a^4*b^3)/(16*(a^8*b - a^9)))^(1/2) - (cos(c + d*x)*(48*a*b^8 + 96*b^9 + 9*
a^2*b^7))/(16*a^4))*(-((a^9*b^5)^(1/2) - a^4*b^3)/(16*(a^8*b - a^9)))^(1/2)*1i - (((768*a^3*b^8 - 144*a^5*b^6)
/(64*a^5) + (((10240*a^8*b^5 - 12288*a^7*b^6 + 6144*a^9*b^4)/(64*a^5) + (cos(c + d*x)*(12288*a^8*b^5 - 8192*a^
9*b^4)*(-((a^9*b^5)^(1/2) - a^4*b^3)/(16*(a^8*b - a^9)))^(1/2))/(16*a^4))*(-((a^9*b^5)^(1/2) - a^4*b^3)/(16*(a
^8*b - a^9)))^(1/2) - (cos(c + d*x)*(2304*a^4*b^7 + 768*a^5*b^6 + 144*a^6*b^5))/(16*a^4))*(-((a^9*b^5)^(1/2) -
a^4*b^3)/(16*(a^8*b - a^9)))^(1/2))*(-((a^9*b^5)^(1/2) - a^4*b^3)/(16*(a^8*b - a^9)))^(1/2) + (cos(c + d*x)*(
48*a*b^8 + 96*b^9 + 9*a^2*b^7))/(16*a^4))*(-((a^9*b^5)^(1/2) - a^4*b^3)/(16*(a^8*b - a^9)))^(1/2)*1i)/((((768*
a^3*b^8 - 144*a^5*b^6)/(64*a^5) + (((10240*a^8*b^5 - 12288*a^7*b^6 + 6144*a^9*b^4)/(64*a^5) - (cos(c + d*x)*(1
2288*a^8*b^5 - 8192*a^9*b^4)*(-((a^9*b^5)^(1/2) - a^4*b^3)/(16*(a^8*b - a^9)))^(1/2))/(16*a^4))*(-((a^9*b^5)^(
1/2) - a^4*b^3)/(16*(a^8*b - a^9)))^(1/2) + (cos(c + d*x)*(2304*a^4*b^7 + 768*a^5*b^6 + 144*a^6*b^5))/(16*a^4)
)*(-((a^9*b^5)^(1/2) - a^4*b^3)/(16*(a^8*b - a^9)))^(1/2))*(-((a^9*b^5)^(1/2) - a^4*b^3)/(16*(a^8*b - a^9)))^(
1/2) - (cos(c + d*x)*(48*a*b^8 + 96*b^9 + 9*a^2*b^7))/(16*a^4))*(-((a^9*b^5)^(1/2) - a^4*b^3)/(16*(a^8*b - a^9
)))^(1/2) + (((768*a^3*b^8 - 144*a^5*b^6)/(64*a^5) + (((10240*a^8*b^5 - 12288*a^7*b^6 + 6144*a^9*b^4)/(64*a^5)
+ (cos(c + d*x)*(12288*a^8*b^5 - 8192*a^9*b^4)*(-((a^9*b^5)^(1/2) - a^4*b^3)/(16*(a^8*b - a^9)))^(1/2))/(16*a
^4))*(-((a^9*b^5)^(1/2) - a^4*b^3)/(16*(a^8*b - a^9)))^(1/2) - (cos(c + d*x)*(2304*a^4*b^7 + 768*a^5*b^6 + 144
*a^6*b^5))/(16*a^4))*(-((a^9*b^5)^(1/2) - a^4*b^3)/(16*(a^8*b - a^9)))^(1/2))*(-((a^9*b^5)^(1/2) - a^4*b^3)/(1
6*(a^8*b - a^9)))^(1/2) + (cos(c + d*x)*(48*a*b^8 + 96*b^9 + 9*a^2*b^7))/(16*a^4))*(-((a^9*b^5)^(1/2) - a^4*b^
3)/(16*(a^8*b - a^9)))^(1/2) + (9*a*b^8 + 24*b^9)/(32*a^5)))*(-((a^9*b^5)^(1/2) - a^4*b^3)/(16*(a^8*b - a^9)))
^(1/2)*2i)/d - ((5*cos(c + d*x))/(8*a) - (3*cos(c + d*x)^3)/(8*a))/(d*(cos(c + d*x)^4 - cos(c + d*x)^2 + sin(c
+ d*x)^2)) - (atan(((((cos(c + d*x)*(48*a*b^8 + 96*b^9 + 9*a^2*b^7))/(16*a^4) - (((12*a^3*b^8 - (9*a^5*b^6)/4
)/a^5 + ((3*a + 8*b)*((((160*a^8*b^5 - 192*a^7*b^6 + 96*a^9*b^4)/a^5 - (cos(c + d*x)*(12288*a^8*b^5 - 8192*a^9
*b^4)*(3*a + 8*b))/(256*a^6))*(3*a + 8*b))/(16*a^2) + (cos(c + d*x)*(2304*a^4*b^7 + 768*a^5*b^6 + 144*a^6*b^5)
)/(16*a^4)))/(16*a^2))*(3*a + 8*b))/(16*a^2))*(3*a + 8*b)*1i)/(16*a^2) + (((cos(c + d*x)*(48*a*b^8 + 96*b^9 +
9*a^2*b^7))/(16*a^4) + (((12*a^3*b^8 - (9*a^5*b^6)/4)/a^5 + ((3*a + 8*b)*((((160*a^8*b^5 - 192*a^7*b^6 + 96*a^
9*b^4)/a^5 + (cos(c + d*x)*(12288*a^8*b^5 - 8192*a^9*b^4)*(3*a + 8*b))/(256*a^6))*(3*a + 8*b))/(16*a^2) - (cos
(c + d*x)*(2304*a^4*b^7 + 768*a^5*b^6 + 144*a^6*b^5))/(16*a^4)))/(16*a^2))*(3*a + 8*b))/(16*a^2))*(3*a + 8*b)*
1i)/(16*a^2))/(((9*a*b^8)/32 + (3*b^9)/4)/a^5 - (((cos(c + d*x)*(48*a*b^8 + 96*b^9 + 9*a^2*b^7))/(16*a^4) - ((
(12*a^3*b^8 - (9*a^5*b^6)/4)/a^5 + ((3*a + 8*b)*((((160*a^8*b^5 - 192*a^7*b^6 + 96*a^9*b^4)/a^5 - (cos(c + d*x
)*(12288*a^8*b^5 - 8192*a^9*b^4)*(3*a + 8*b))/(256*a^6))*(3*a + 8*b))/(16*a^2) + (cos(c + d*x)*(2304*a^4*b^7 +
768*a^5*b^6 + 144*a^6*b^5))/(16*a^4)))/(16*a^2))*(3*a + 8*b))/(16*a^2))*(3*a + 8*b))/(16*a^2) + (((cos(c + d*
x)*(48*a*b^8 + 96*b^9 + 9*a^2*b^7))/(16*a^4) + (((12*a^3*b^8 - (9*a^5*b^6)/4)/a^5 + ((3*a + 8*b)*((((160*a^8*b
^5 - 192*a^7*b^6 + 96*a^9*b^4)/a^5 + (cos(c + d*x)*(12288*a^8*b^5 - 8192*a^9*b^4)*(3*a + 8*b))/(256*a^6))*(3*a
+ 8*b))/(16*a^2) - (cos(c + d*x)*(2304*a^4*b^7 + 768*a^5*b^6 + 144*a^6*b^5))/(16*a^4)))/(16*a^2))*(3*a + 8*b)
)/(16*a^2))*(3*a + 8*b))/(16*a^2)))*(3*a + 8*b)*1i)/(8*a^2*d)